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# Cubic Special Primes

Cubic Special Primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

n   is smallest prime such that the difference of successive terms are the smallest cubics of positive integers, where     n   <   15000.

## ALGOL 68

BEGIN # find a sequence of primes where the members differ by a cube #
INT max prime = 15 000;
# sieve the primes to max prime #
[]BOOL prime = PRIMESIEVE max prime;
# construct a table of cubes, we will need at most the cube root of max prime #
[ 1 : ENTIER exp( ln( max prime ) / 3 ) ]INT cube;
FOR i TO UPB cube DO cube[ i ] := i * i * i OD;
# find the prime sequence #
print( ( "2" ) );
INT p := 2;
WHILE p < max prime DO
# find a prime that is p + a cube #
INT q := 0;
FOR i WHILE q := p + cube[ i ];
IF q > max prime THEN FALSE ELSE NOT prime[ q ] FI
DO SKIP OD;
IF q <= max prime THEN print( ( " ", whole( q, 0 ) ) ) FI;
p := q
OD;
print( ( newline ) )
END
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## ALGOL W

begin % find a sequence of primes where the members differ by a cube %
integer MAX_PRIME, MAX_CUBE;
MAX_PRIME := 15000;
MAX_CUBE  := entier( exp( ln( MAX_PRIME ) / 3 ) );
begin
logical array prime ( 1 :: MAX_PRIME );
integer array cube ( 1 :: MAX_CUBE );
integer p, q, c;
% sieve the primes to MAX_PRIME %
prime( 1 ) := false; prime( 2 ) := true;
for i := 3 step 2 until MAX_PRIME do prime( i ) := true;
for i := 4 step 2 until MAX_PRIME do prime( i ) := false;
for i := 3 step 2 until entier( sqrt( MAX_PRIME ) ) do begin
integer ii; ii := i + i;
if prime( i ) then for s := i * i step ii until MAX_PRIME do prime( s ) := false
end for_i;
% construct a table of cubes, we will need at most the cube root of max prime %
for i := 1 until MAX_CUBE do cube( i ) := i * i * i;
% find the prime sequence %
writeon( "2" );
p := 2;
while p < MAX_PRIME do begin
% find a prime that is p + a cube %
c := 1;
q := p + cube( c );
while if q > MAX_PRIME then false else not prime( q ) do begin
c := c + 1;
q := p + cube( c )
end while_not_prime_q ;
if q <= MAX_PRIME then writeon( i_w := 1, s_w := 0, " ", q );
p := q
end while_p_lt_MAX_PRIME
end;
write()
end.
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## AWK

# syntax: GAWK -f CUBIC_SPECIAL_PRIMES.AWK
# converted from FreeBASIC
BEGIN {
start = p = 2
stop = 15000
n = 1
printf("%5d ",p)
count = 1
do {
if (is_prime(p + n^3)) {
p += n^3
n = 1
printf("%5d%1s",p,++count%10?"":"\n")
}
else {
n++
}
} while (p + n^3 <= stop)
printf("\nCubic special primes %d-%d: %d\n",start,stop,count)
exit(0)
}
function is_prime(x, i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
}

Output:
2     3    11    19    83  1811  2027  2243  2251  2467
2531  2539  3539  3547  4547  5059 10891 12619 13619 13627
13691 13907 14419
Cubic special primes 2-15000: 23

## F#

// Cubic Special Primes: Nigel Galloway. March 30th., 2021
let fN=let N=[for n in [0..25]->n*n*n] in let mutable n=2 in (fun g->match List.contains(g-n)N with true->n<-g; true |_->false)
primes32()|>Seq.takeWhile((>)16000)|>Seq.filter fN|>Seq.iter(printf "%d "); printfn ""

Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## Factor

Works with: Factor version 0.98
USING: fry io kernel lists lists.lazy math math.functions
math.primes prettyprint ;

2 [ 1 lfrom swap '[ 3 ^ _ + ] lmap-lazy [ prime? ] lfilter car ]
lfrom-by [ 15000 < ] lwhile [ pprint bl ] leach nl
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## FreeBASIC

#include once "isprime.bas"

dim as integer p = 2, n = 1

print 2;" ";

do
if isprime(p + n^3) then
p += n^3
n = 1
print p;" ";
else
n += 1
end if
loop until p + n^3 >= 15000
print
end
Output:

2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## Go

Translation of: Wren
package main

import (
"fmt"
"math"
)

func sieve(limit int) []bool {
limit++
// True denotes composite, false denotes prime.
c := make([]bool, limit) // all false by default
c[0] = true
c[1] = true
// no need to bother with even numbers over 2 for this task
p := 3 // Start from 3.
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}

func isCube(n int) bool {
s := int(math.Cbrt(float64(n)))
return s*s*s == n
}

func commas(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}

func main() {
c := sieve(14999)
fmt.Println("Cubic special primes under 15,000:")
fmt.Println(" Prime1 Prime2 Gap Cbrt")
lastCubicSpecial := 3
gap := 1
count := 1
fmt.Printf("%7d %7d %6d %4d\n", 2, 3, 1, 1)
for i := 5; i < 15000; i += 2 {
if c[i] {
continue
}
gap = i - lastCubicSpecial
if isCube(gap) {
cbrt := int(math.Cbrt(float64(gap)))
fmt.Printf("%7s %7s %6s %4d\n", commas(lastCubicSpecial), commas(i), commas(gap), cbrt)
lastCubicSpecial = i
count++
}
}
fmt.Println("\n", count+1, "such primes found.")
}
Output:
Same as Wren example.

## jq

Works with: jq

Works with gojq, the Go implementation of jq

For the definition of `is_prime` used here, see https://rosettacode.org/wiki/Additive_primes
# Output: an array
def cubic_special_primes:
. as \$N
| sqrt as \$maxidx
| {cprimes: [2], q: 0}
| until( .cprimes[-1] >= \$N or .q >= \$N;
label \$out
| foreach range(1; \$maxidx + 1) as \$i (.;
.q = (.cprimes[-1] + (\$i * \$i * \$i))
| if .q >= \$N
then .emit = true
elif .q | is_prime
then .cprimes = .cprimes + [.q]
| .emit = true
else .
end;
select(.emit)) | {cprimes, q}, break \$out )
| .cprimes ;

15000
| ("Cubic special primes < \(.):",
cubic_special_primes)
Output:
[2,3,11,19,83,1811,2027,2243,2251,2467,2531,2539,3539,3547,4547,5059,10891,12619,13619,13627,13691,13907,14419]

## Julia

using Primes

function cubicspecialprimes(N = 15000)
cprimes, maxidx = [2], isqrt(N)
while (n = cprimes[end]) < N
for i in 1:maxidx
q = n + i * i * i
if q > N
return cprimes
elseif pmask[q] # got next qprime
push!(cprimes, q)
break
end
end
end
end

println("Cubic special primes < 15000:")
foreach(p -> print(rpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""), enumerate(cubicspecialprimes()))

Output:
Cubic special primes < 16000:
2     3     11    19    83    1811  2027  2243  2251  2467
2531  2539  3539  3547  4547  5059  10891 12619 13619 13627
13691 13907 14419

## Mathematica/Wolfram Language

start = {2};
Do[
If[PrimeQ[i],
last = Last[start];
If[IntegerQ[Surd[i - last, 3]],
AppendTo[start, i]
]
]
,
{i, 3, 15000}
]
start
Output:
{2,3,11,19,83,1811,2027,2243,2251,2467,2531,2539,3539,3547,4547,5059,10891,12619,13619,13627,13691,13907,14419}

## Nim

Translation of: Go
import math, strformat

func sieve(limit: Positive): seq[bool] =
# True denotes composite, false denotes prime.
result = newSeq[bool](limit + 1) # All false by default.
result[0] = true
result[1] = true
# No need to bother with even numbers over 2 for this task.
var p = 3
while true:
let p2 = p * p
if p2 > limit: break
for i in countup(p2, limit, 2 * p):
result[i] = true
while true:
inc p, 2
if not result[p]: break

func isCube(n: int): bool =
let s = cbrt(n.toFloat).int
result = s * s * s == n

let c = sieve(14999)
echo "Cubic special primes under 15_000:"
echo " Prime1 Prime2 Gap Cbrt"
var lastCubicSpecial = 3
var count = 1
echo &"{2:7} {3:7} {1:6} {1:4}"
for n in countup(5, 14999, 2):
if c[n]: continue
let gap = n - lastCubicSpecial
if gap.isCube:
let gapCbrt = cbrt(gap.toFloat).int
echo &"{lastCubicSpecial:7} {n:7} {gap:6} {gapCbrt:4}"
lastCubicSpecial = n
inc count
echo &"\n{count + 1} such primes found."
Output:
Cubic special primes under 15_000:
Prime1  Prime2    Gap  Cbrt
2       3      1    1
3      11      8    2
11      19      8    2
19      83     64    4
83    1811   1728   12
1811    2027    216    6
2027    2243    216    6
2243    2251      8    2
2251    2467    216    6
2467    2531     64    4
2531    2539      8    2
2539    3539   1000   10
3539    3547      8    2
3547    4547   1000   10
4547    5059    512    8
5059   10891   5832   18
10891   12619   1728   12
12619   13619   1000   10
13619   13627      8    2
13627   13691     64    4
13691   13907    216    6
13907   14419    512    8

23 such primes found.

## Perl

Library: ntheory
use strict;
use warnings;
use feature 'say';
use ntheory 'is_prime';

my @sp = my \$previous = 2;
do {
my(\$next,\$n);
while () { last if is_prime( \$next = \$previous + ++\$n**3 ) }
push @sp, \$next;
\$previous = \$next;
} until \$sp[-1] >= 15000;

pop @sp and say join ' ', @sp;
Output:
2 3 11 19 83 1811 2027 2243 2251 2467 2531 2539 3539 3547 4547 5059 10891 12619 13619 13627 13691 13907 14419

## Raku

A two character difference from the Quadrat Special Primes entry. (And it could have been one.)

my @sqp = 2, -> \$previous {
my \$next;
for (1..).map: *³ {
\$next = \$previous + \$_;
last if \$next.is-prime;
}
\$next
}*;

say "{+\$_} matching numbers:\n", \$_».fmt('%5d').batch(7).join: "\n" given
@sqp[^(@sqp.first: * > 15000, :k)];
Output:
23 matching numbers:
2     3    11    19    83  1811  2027
2243  2251  2467  2531  2539  3539  3547
4547  5059 10891 12619 13619 13627 13691
13907 14419

## REXX

/*REXX program finds the smallest primes such that the difference of successive terms   */
/*───────────────────────────────────────────────────── are the smallest cubic numbers. */
parse arg hi cols . /*obtain optional argument from the CL.*/
if hi=='' | hi=="," then hi= 15000 /* " " " " " " */
if cols=='' | cols=="," then cols= 10 /* " " " " " " */
call genP /*build array of semaphores for primes.*/
w= 10 /*width of a number in any column. */
title= 'the smallest primes < ' commas(hi) " such that the" ,
'difference of successive terma are the smallest cubic numbers'
if cols>0 then say ' index │'center(title, 1 + cols*(w+1) ) /*display the title.*/
if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') /* " " sep.*/
found= 0; idx= 1 /*initialize number of cbp and index.*/
op= 1
\$= /*a list of cubic primes (so far). */
do j=0 by 0
do k=1 until !.np
np= op + k**3 /*find the next cube plus the oldPrime.*/
if np>= hi then leave j /*Is newPrime too big? Then quit. */
end /*k*/
found= found + 1 /*bump number of primes of this type. */
op= np /*assign the newPrime to the oldPrime*/
if cols<0 then iterate /*Build the list (to be shown later)? */
c= commas(np) /*maybe add commas to the number. */
\$= \$ right(c, max(w, length(c) ) ) /*add a cubic prime──►list, allow big#.*/
if found//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr(\$, 2); \$= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/

if \$\=='' then say center(idx, 7)"│" substr(\$, 2) /*possible display residual output.*/
if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─') /*display foot sep. */
say
say 'Found ' commas(found) " of " title
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: !.= 0 /*placeholders for primes (semaphores).*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
!.2=1;  !.3=1;  !.5=1;  !.7=1;  !.11=1 /* " " " " flags. */
#=5; s.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do [email protected].#+2 by 2 to hi /*find odd primes from here on. */
parse var j '' -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; s.#= j*j;  !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return
output   when using the default inputs:
index │  the smallest primes  <  15,000  such that the difference of successive terma are the smallest cubic numbers
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
1   │          2          3         11         19         83      1,811      2,027      2,243      2,251      2,467
11   │      2,531      2,539      3,539      3,547      4,547      5,059     10,891     12,619     13,619     13,627
21   │     13,691     13,907     14,419
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  23  of  the smallest primes  <  15,000  such that the difference of successive terma are the smallest cubic numbers

## Ring

see "working..." + nl

Primes = []
limit1 = 50
oldPrime = 2

for n = 1 to limit1
nextPrime = oldPrime + pow(n,3)
if isprime(nextPrime)
n = 1
oldPrime = nextPrime
else
nextPrime = nextPrime - oldPrime
ok
next

see "prime1 prime2 Gap Cbrt" + nl
for n = 1 to Len(Primes)-1
diff = Primes[n+1] - Primes[n]
for m = 1 to diff
if pow(m,3) = diff
cbrt = m
exit
ok
next
see ""+ Primes[n] + " " + Primes[n+1] + " " + diff + " " + cbrt + nl
next

see "Found " + Len(Primes) + " of the smallest primes < 15,000 such that the difference of successive terma are the smallest cubic numbers" + nl

see "done..." + nl

Output:
working...
prime1 prime2 Gap Cbrt
2      3    1     1
3      11    8     2
11      19    8     2
19      83    64     4
83      1811    1728     12
1811      2027    216     6
2027      2243    216     6
2243      2251    8     2
2251      2467    216     6
2467      2531    64     4
2531      2539    8     2
2539      3539    1000     10
3539      3547    8     2
3547      4547    1000     10
4547      5059    512     8
5059      10891    5832     18
10891      12619    1728     12
12619      13619    1000     10
13619      13627    8     2
13627      13691    64     4
13691      13907    216     6
13907      14419    512     8
Found 23 of the smallest primes < 15,000  such that the difference of successive terma are the smallest cubic numbers
done...

## Sidef

func cubic_primes(callback) {

var prev = 2
callback(prev)

loop {
var curr = (1..Inf -> lazy.map { prev + _**3 }.first { .is_prime })
callback(curr)
prev = curr
}
}

say gather {
cubic_primes({|k|
break if (k >= 15000)
take(k)
})
}
Output:
[2, 3, 11, 19, 83, 1811, 2027, 2243, 2251, 2467, 2531, 2539, 3539, 3547, 4547, 5059, 10891, 12619, 13619, 13627, 13691, 13907, 14419]

## Wren

Library: Wren-math
Library: Wren-fmt
import "/math" for Int, Math
import "/fmt" for Fmt

var isCube = Fn.new { |n|
var c = Math.cbrt(n).round
return c*c*c == n
}

var primes = Int.primeSieve(14999)
System.print("Cubic special primes under 15,000:")
System.print(" Prime1 Prime2 Gap Cbrt")
var lastCubicSpecial = 3
var gap = 1
var count = 1
Fmt.print("\$,7d \$,7d \$,6d \$4d", 2, 3, 1, 1)
for (p in primes.skip(2)) {
gap = p - lastCubicSpecial
if (isCube.call(gap)) {
Fmt.print("\$,7d \$,7d \$,6d \$4d", lastCubicSpecial, p, gap, Math.cbrt(gap).round)
lastCubicSpecial = p
count = count + 1
}
}
System.print("\n%(count+1) such primes found.")
Output:
Cubic special primes under 15,000:
Prime1  Prime2    Gap  Cbrt
2       3      1    1
3      11      8    2
11      19      8    2
19      83     64    4
83   1,811  1,728   12
1,811   2,027    216    6
2,027   2,243    216    6
2,243   2,251      8    2
2,251   2,467    216    6
2,467   2,531     64    4
2,531   2,539      8    2
2,539   3,539  1,000   10
3,539   3,547      8    2
3,547   4,547  1,000   10
4,547   5,059    512    8
5,059  10,891  5,832   18
10,891  12,619  1,728   12
12,619  13,619  1,000   10
13,619  13,627      8    2
13,627  13,691     64    4
13,691  13,907    216    6
13,907  14,419    512    8

23 such primes found.